statementsmeasurable

Statementsmeasurable is a term used in probability theory and measure theory to describe a statement about outcomes of a random experiment that is measurable with respect to a given sigma-algebra. In practical terms, a statement is considered measurable if the set of outcomes for which the statement holds is an element of the sigma-algebra, making its probability well-defined.

Formal definition: Let (Ω, F, P) be a probability space. A statement S about outcomes ω ∈ Ω is measurable with respect to F if the truth set {ω ∈ Ω : S is true} belongs to F. If S is formulated in terms of a random variable X, for example S = {X ∈ A}, then S is measurable whenever X is F-measurable and A is a measurable set (such as a Borel set). Common examples include S = {X > 0} and S = {X ∈ B} for B in the Borel sigma-algebra.

Notes: Non-measurable statements can be constructed in certain abstract spaces using non-measurable sets, in which case the statement does not have a well-defined probability. In applied contexts, most statements of interest are measurable because they are defined via measurable functions or by membership in a sigma-algebra generated by observed data. In the theory of stochastic processes, the concept extends to filtrations, where a statement is considered measurable or adapted with respect to the information available up to a given time. The notion of statementsmeasurable aligns with the broader concept of measurability in measure theory and connects to topics such as random variables, sigma-algebras, and measurable functions.